Question

The period of a function {2^{\left\\{ x \right\\}}} + \sin \pi x + {3^{\left\\{ {\dfrac{x}{2}} \right\\}}} + \cos 2\pi x (where \left\\{ x \right\\}denotes the fractional part of x) is
A.2
B.1
C.3
D.None of these

Answer 1

In this question first of all we calculate the period of all the individual terms of the given function. After that, to calculate the period of a function take Least common multiple (LCM) of period of all the terms. Which will be the period of our given function.

Complete step-by-step answer:
We have given f\left( x \right) = $$$${2^{\left\\{ x \right\\}}} + \sin \pi x + {3^{\left\\{ {\dfrac{x}{2}} \right\\}}} + \cos 2\pi x
As we know that,
Period of fractional part is always 1
$\Rightarrow x = 1$
$\Rightarrow$Period of {2^{\left\\{ x \right\\}}}= 1 $\ldots \left( 1 \right)$
For period of {3^{\left\\{ {\dfrac{x}{2}} \right\\}}}
Consider, $\dfrac{x}{2} = 1$
\Rightarrow $$$$x = 2
$\Rightarrow$Period of {3^{\left\\{ {\dfrac{x}{2}} \right\\}}}=2 $\ldots \left( 2 \right)$
Period of sine function is $2\pi$
$\Rightarrow$ Period of $\sin \pi x$=$2\pi$
$\Rightarrow$ Period of $\pi x = 2\pi$
$\Rightarrow x = \dfrac{{2\pi }}{\pi }$
$\Rightarrow$ Period of $\sin \pi x$ is $x = 2$ $\ldots \left( 3 \right)$
Period of cosine function is $2\pi$
For period of $\cos 2\pi x$
Consider, $2\pi x = 2\pi$
$\Rightarrow x = \dfrac{{2\pi }}{{2\pi }}$
$\Rightarrow x = 1$ $\ldots \left( 4 \right)$
For calculating the period of any function we have to take LCM of the period of all the terms given in the above function.
So, period of {2^{\left\\{ x \right\\}}} + \sin \pi x + {3^{\left\\{ {\dfrac{x}{2}} \right\\}}} + \cos 2\pi x is LCM $\left( {1,2,2,1} \right)$ $\ldots$(from 1,2,3 and 4)
Thus, period of {2^{\left\\{ x \right\\}}} + \sin \pi x + {3^{\left\\{ {\dfrac{x}{2}} \right\\}}} + \cos 2\pi x is 2.
Hence, option A. 2 is the correct option.

Note: Period of a function: The distance between the repetition of any function is called the period of the function.
For a trigonometric function the length of one complete cycle is called a period.
Fraction part function is a periodic function with period 1.